Beyond their role as correlative conjunctions in either/or, neither/nor, both either and neither can function as pronouns:
Cooperation between these two cultures of biomedical research is the only hope for clinical and basic biomedical science, and neither can exist without the other, contrary to what either may wish. — Bradley P. Fuhrman, Jerry J. Zimmerman, Pediatric Critical Care E-Book, 2011, 51.
Both can modify nouns — and even the pronoun one — like adjectives:
Although neither worker observed the FC rules concerning face masks, the medical histories point against any link between the condition of either worker and any exposure to 2,4.5-T [Agent Orange]. Current Status of Agent Orange Studies, 1981.
Morgan had little faith in either one. “Neither one of them has a clue about what's about to happen. Doc, I don't care what it takes. I need to stay.” — John Martin Recovery, 2010, 254.
In standard English, however, only either can function as an adverb, meaning ‘also, as well, to the same degree’ with a preceding negative expression. It comes in clause-final position:
This stance, he admits, suggests that individuals could, in principle, not really live aesthetically, but yet not live ethically either. — Robert L. Perkins, ed., International Kierkegaard Commentary: Either/or, Part 2, 2003, 160.
In non-standard dialects or colloquial speech that permits double negation, neither can be used in the same fashion:
“Mama says she don't need her worthless brothers, so we don't, neither.” — Marilyn Pappano, Undercover in Copper Lake, 2014, 61.